Dividing by r^4 and writing Y = y/r gives: Y^4 - 8Y^3 - 8Y + n/r^4 = 0(1)For fixed n and r, this quartic in Y (has one stationary point - a minimum value of n/r^4 - 480.219..when Y = 6.0545...(both irrationals). When n = 0 it has roots at Y = 0 and 8.121..It follows that, for n > 0, there will be two real solutions for Y provided that n/r^4 < 480.219.. (otherwise none) and thatthese solutions will be in the range 0 < Y < 8.121..Now, we are looking for rational solutions for Y, so that y and r can be integers, and we can now say:For a given r value, there will be, at most, two integer values for y and they will satisfy the inequalitiesy < 8r and n < 480.219 r^4.

This is still very vague, but these limits allow computer searches to be better organized and more conclusive, which, in turn, gave me an insight into the following..